computer search for defects in a d = 3 heisenberg spin glass · some of my approach was inspired by...

ANNALS OF PHYSICS 156, 324-367 (1984)

Computer Search for Defects in a D = 3 Heisenberg Spin Glass

CHRISTOPHER L. HENLEY *

Department of Physics, Harvard University, Cambridge, Massachusetts 02138 *

Received August 26, 1983; revised November 7, 1983

A computer study of T= 0 metastable states (local minima) of a Heisenberg spin glass (1728 fixed-length spins on a simple cubic latice with random nearest-neighbor bonds) is reported, in which an attempt is made to describe the relation of nearby minima in terms of defects and to characterize those defects. The local rotation matrix relating one equilibrium configuration to another is well defined, and has a (relatively long) correlation length of 34 lattice units. The size of the clusters of spins which tend to be locked together in rigid rotations is 100-200 spins by several criteria. It is found that disclination lines and continuous twists of 360” across the sample exist as stable minima of the energy. However, the domain walls, across which the rotation matrix undergoes a reflection, appear to be more “fundamental” defects. The excitation energy of a defect is of the order of one exchange coupling or less. It does not seem possible to resolve the differences between two metastable states as a simple combination of spatially isolated defects.

I. INTRODUCTION

Of the many computer simulations of spin glasses, relatively few [l-5] have aimed at visualizing the nature of the spin configurations which are local minima of the energy (which I will call EC’s, for “equilibrium configurations”) and the relationship between minima which are nearby in the 2N-dimensional configuration space. Since the EC’s are analogs for the spin glass of the ground state for an ordered system, we need to understand them in order to model the low-temperature behavior of spin glasses. Only some of these simulations [l-3] used vector spins with finite range interactions, the most realistic case. In this paper I report computer experiments in which, first, EC’s of Heisenberg (m = 3) spins on a three-dimensional lattice were obtained; then, for each pair of EC’s the SO(3) rotation matrices relating the two were calculated. The resulting configurations of slowly varying SO(3) matrices were analyzed in hopes of identifying topological defects or textures. Here I mean defects in the relative sense, as discussed in Section III of the preceding paper [6]: rather than comparing spin glass EC’s to some ordered configuration (which obviously gives defects everywhere), one compares different EC’s with each other. The possible

* Current address: AT&T Bell Laboratories, ID-263, Murray Hill, N. J., 07974.

324 0003.4916184 $7.50 Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.

SEARCH FOR DEFECTS IN A SPIN GLASS 325

types of topological defects, originally classified by Toulouse [7], were discussed in the same place [6].

Some previous authors have simulated spin glasses on the computer to look for defects. Reed [ 11 investigated how large a cluster must be to give a new metastable state after it is rotated and relaxed. In their study of the EC’s of XY spin glasses, Grzonka and Moore [3] present a picture of a pair of relative vortices (in d = 2) produced by randomizing the spins in a small cluster, subsequently relaxing them, and comparing to the original EC. A number of studies have been done of the d = 2 Ising model on a square lattice with nearest neighbor J = f 1, the “frustration model” [4, 51. Some of my approach was inspired by work on dislocations in metallic glasses [8]. Looking for topological defects in computer simulations of Heisenberg spin glasses has also been suggested by Toulouse [7].

I take the usual exchange Hamiltonian,

with J, = Jji. For this simulation I have taken the Edwards-Anderson model of the exchange couplings: the spin sites Zi form a simple cubic lattice and the coupling J, is a Gaussian random variable of unit variance for nearest-neighbor pairs (i,j), otherwise zero. There are N = L3 = 123 = 1728 spins, which are three-component unit vectors. (N= 1728 is actually not a very large system in the Heisenberg case; I estimate it has the same number of distinct local minima as an Ising spin glass with only Ng 70 spins.)

Limited tests of the model with nearest neighbor J, = f 1 gave similar results. The algorithms described below were also implemented for the RKKY couplings which model the &Mn spin glass [9]. Since, in the RKKY case, each spin is coupled to many neighbor spins, it is much costlier in computer time to generate the EC’s and this was not pursued. Since both Edwards-Anderson and RKKY EC’s certainly have the same global rotation symmetry and since spin stiffness calculations [ 10-121 indicate they have similar responses when a slowly varying twist of the spins is imposed, one expects that qualitatively identical results would be obtained for RKKY systems.

There are two conditions for a configuration to be an EC. First, the first derivative of the energy with respect to each spin must be zero, which gives

(1.2)

where ii is known as the local exchange field. Second, the Hessian (the second derivative matrix, discussed in Appendix A, where it is called E) must be nonnegative definite. It is characteristic of a spin glass that there are many EC’s.

Below, in Section II, I first discuss the relaxation algorithm for generating EC’s and its problems of convergence-a matter of some interest, since this algorithm is widely used [ 1, 3, 9-131. Some details of the theory of the relaxation are removed to

326 CHRISTOPHERL.HENLEY

Appendix A. Next, in Section III, I describe a procedure for comparison of two EC’s which I used in each experiment. (To understand the statistics of the results, I include a calculation based on random matrix theory in Appendix B.) Once a given realization with a specific set of couplings Jii was produced, the ideal experiment would have been to generate arbitrary, independent EC’s, to compare them, and to look for defects. But the apparent density of defects was too high in this case to give any clear picture. Instead, three kinds of experiments were feasible: (a) to generate arbitrary, independent EC’s and calculate correlation functions, (b) to insert just one defect “by hand” and then relax the EC again, the object being to see if pairs of stable configurations existed which could be described as differing by one defect, and (c) to generate arbitrary, independent EC’s but in smaller samples which were essentially one or two dimensional in shape, the hope being that “natural” defects would be easier to see in this case. The results of these experiments are reported in Sections IV, V, and VI, respectively, along with discussions of the importance of the “reflection” defect. In Section VII, I summarize the results and implications, and gather estimates of the typical cluster or “patch” size N,. Due to limitations on computer time, many of the results must be considered tentative.

II. RELAXATION PROCEDURE AND EQUILIBRIUM CONFIGURATIONS

In this section I discuss the procedures used (1) to produce a particular realization of the exchange couplings for a sample of spin glass, and (2) to generate EC’s of the sample by relaxation. I discuss the convergence problems of the relaxation in some detail (with more detail in the Appendix) and note some results on the distribution of the energies of the EC’s and of the local exchange fields on the spins within them.

The 3N exchange couplings J, were chosen as independent Gaussian random variables of zero mean and variance ? = 1. However, statistical fluctuations in the average variance for a finite4 sample, zrr = (3N): CicjJi, lead to unwanted fluctuations in the energy of EC’s (the variance of Jerr is (6N))‘). Therefore, in all energies reported here, I have normalized the couplings replacing J, + Jij/Jefeff, which is equivalent to using a distribution proportional to S(Ci,j Jb - 3N).

To generate an EC, I started from a configuration of independent, randomly oriented unit spins, which was then relaxed using the Walker-Walstedt procedure [q]. Each of the N spins is updated once per iteration. In the update of spin i, we set 6 to the unit vector parallel to the instantaneous local field &; the new value of Zi is independent of the old value (except indirectly, through neighboring spins which have been reoriented since the previous time that Zi was changed.) The sequence of spins for the updates is generated at the beginning with a random number generator, but for reproducibility the same ordering was used in each succeeding iteration. Note that the energy strictly decreases in each update unless we are already at a solution of (1.2), i.e., an EC; since the energy is bounded below we are sure to converge to a minimum eventually. (If we start at certain saddle points of the energy function, the algorithm


will stay there, but the set of starting configurations that will go to a saddle point has measure zero.)

This procedure differs from true Monte Carlo simulations in which there is an external heat bath at a finite temperature, implemented by randomness in the iterations. From that point of view, the procedure I used consists of (i) heating the spins to infinite temperature, and (ii) instantaneously cooling the bath to zero temperature. It would not be surprising, then, if such a quench trapped the system in relatively high metastable states; as we shall see, it does.

A. Convergence of the Relaxation Process

A priori, we do not know the energy of the minimum to which we are converging. Thus, the only indications we can have that we are approaching a minimum is that the spins are better aligned with their local fields, or (what turns out to be equivalent) that they change more and more slowly. We will see that, unfortunately, these indications can be false.

I define three quantities which may be used as measures of convergence:

D = $x ($ - &/hi)*, I

W=;z(hi-h;GJ, I

(2.2)

and (taking one iteration as the time unit)

1 dE/dt 1 = E(t) - E(t + 1). (2.3)

So D is just the average squared deviation of a spin from being parallel to its instantaneous exchange field, and W is a similar average, but weighted by h,/2. Note that W is related to D just as the energy difference E(t) -E, is related to xi I< - ?i”‘l*, the squared “distance” in configuration space from the minimum; thus one imagines that W might be an estimate of E(t) -E, or at least a convergence measure independent of D. However, D, W, and IdE/dtI are essentially proportional to each other as is shown in the second portion of the Appendix.

I also defined D’ and W’ by (2.1) and (2.2), but in a running sum in which the ith term was evaluated just before the ith spin was updated. Just after that term is zero, so D’ z 20, W’ g 2 W. The observed D’, W’, and / dE/dtI have the predicted ratios (with fluctuations of ~10%).

Because of computer time limitations, the stopping criterion chosen represents an unhappy trade-off between the objectives of (i) relaxing each EC completely, and (ii) generating sufficiently many EC’s to get good statistics. For samples of 123 spins, as used for most of this work, the relaxation continued until D’ was less than lo-‘, or was stopped after 1300 iterations, whichever came first; typically almost 1000 iterations were needed. An additional 50 iterations were performed a check.

328 CHRISTOPHER L. HENLEY

For comparison, using the same algorithm for the same nearest-beighbor interactions, Reed [ 1 l] took “many thousands” of iterations for systems of this size, and one infers that Banavar and Cieplak [ 121 would take about 4500. My choice to relax less than they did was motivated by the belief that the chief object of my simulation, the topological defects, would not change during the later iterations. I assumed that, within 1000 iterations, the system would be sufficiently close to an EC that the energy could be represented by a positive-definite quadratic form in the deviations of the spins from the EC. Once one is in this quadratic valley, the remainder of the relaxation to the minimum is very slow because the energy function (of a spin glass with many degrees of freedom) is very flat in some directions in configuration space [3, 13). Spatially, the modes which relax in these last stages are spin waves or other extended modes with slow spatial variation [9]. Thus no additional topological defects should be created, and it seemed possible to sacrifice complete convergence. In fact, I believe this reasoning is qualitatively correct: certainly inspection of topological defects founds by the comparison algorithm is a much better way to tell whether two configurations are converging to the same EC than direct comparison of the spins, which requires very well relaxed samples.

Unfortunately, detailed observation of the relaxation process suggested that the quadratic valley was rarely reached in 1000 iterations. Typically, after D' had decreased to about 10m5, it no longer decreased monotonically, but oscillated in irregular waves at intervals of 100-300 iterations; see Fig. 1. As I have mentioned,

.

E : ;I05 o,

FIG. 1. Convergence of a relaxation which was begun from a random configuration. The dashed curve is the convergence measure D’ g 20 (on a logarithmic scale) and shows three peaks; these correspond to downward steps of the energy E, which is shown by the solid curve. (The energy unit, in all figures, is the r.m.s. exchange coupling 1) Although some EC’s had as many as six such falls, the fvpical relaxation from random spins had only two falls and stopped about 200 iterations sooner. The horizontal axis begins with the 500th iteration because D’ and E were logged only after this time. One can note that at t = 0, with random spins, D = 2 and E = 0 on the average; after about ten iterations most of the difference E-E, has been relaxed away and it continues to decrease rapidly and monotonically for at least a hundred iterations.


(&/&I was proportional to D’ (with a lag of a few iterations) so that each of the intermittent oscillations corresponds to a relatively sudden “fall” of the energy. There was some tendency for the successive peak D values to get smaller, until D decreased somewhat below lo-‘, after which it usually decayed rather rapidly (with a time constant of the order of 100) to the level of roundoff error, indicating convergence. It is very difficult to predict how long it would oscillate before converging to a true minimum; the time to converge, which is strongly correlated with the number of “falls,” depends strongly on the random starting spin configuration. I will call the relaxation “nominally converged” if D’ < 10e6 and D’ is decreasing during all the last 50 iterations.

It is conceivable that the oscillations are due to spiraling along an elliptical path in configuration space, which is possible with this relaxation algorithm (see Appendix A, following Eq. (A. 17)). However, I believe that the oscillations result from passages near an “almost-minimum,” a saddle point (or perhaps an inflection point) of the energy which has (say) only one unstable eigenvector, the other eigenvalues being distributed as if it were a minimum. Thus, for some time we see relaxation as to a minimum, but eventually the growth of the unstable component takes over and takes us to a new minimum or (more likely) almost-minimum, towards which we again relax. I conjecture that each “fall” of the energy is associated with a single localized defect or two-level system (TLS) of some sort (see discussion in Section VI of the preceding paper[6]). The TLS is in the upper well of its asymmetric two-well potential, which evolves slowly as other parts of the system change; these may “tilt” the upper well higher until the barrier goes to zero, at which point the TLS rapidly falls to the other well. Once the system has reached the lower state, that will change the “tilts” felt by other TLS’s, perhaps enough to make one of them unstable in its turn. This picture has some support from one test run where “snapshots” of the spins before and after two major falls showed little changes except for two patches that underwent a strong “reflection,” which I believe may be the signature of a TLS (see below, Section V, and following sections). One might also note that the average drop of the energy per fall, which is close to 1 unit, is about the same as the estimated defect excitation energy (see under (C), below, and in Section V).

The typical time constant of 100 iterations can be explained in terms of spin waves. As shown in Appendix A, the relaxation rate of a relaxational mode should be its eigenvalue A. If the slow modes are ideal spin waves, then there are nine degenerate slowest modes of wavevector qmi, = 27ilL and &,i, =p,q$” r 0.04. Considering the couplings between the spin waves, which mix different rotation axes and wavevectors and break the degeneracy, it is not surprising to find the observed Amin z 0.01.

B. Energy of EC’s and Local Field Distribution

To check how well the energy has been minimized, one can compare it with the results of previous simulations using the same distribution of couplings. In place of E, I take E = E/N, the energy per spin, for each EC. For the 123-spin EC’s, I found


(E) = -1.846, which is consistent with Huber and Ching’s [ 141 value (E) = -1.84. Reed’s result [ 1 ] (E) = -1.87 may be different because of his use of free boundary conditions, or perhaps because of the statistical fluctuations of J,, (which he did not normalize to 1.)

In view of Section VI of the preceding paper [6], it is interesting to discuss the variation of E = E/N between different EC’s. The standard deviation among dSfSerent realizations was

((&2)J- (&)y* E 5 x 10-3(IEI); (2.4)

this should scale as N- I’* If I omitted the normalization of Jff, then (2.4) would be . about twice as large. The standard deviation of the energy among different EC’s of the same realization was

(5 cc = ((E2)c - (&);)“* z 5 x 10P4(l&l) (2.5)

(the notations (. . .)= for average over spin configurations and (. e e)J for averaging over realizations of the couplings are taken from Section V of Ref. [6]). In comparison, Walker and Walstedt found (TV z 3 x 10e4 with the RKKY spin glass [9, 13).

To understand exactly which EC’s we are finding, it is helpful to consider the entropy-of-minima function g(E). Its definition (see Ref. [ 151 and Section III of Ref. [6]) is that the number Nmin(c) of EC’s of energy per spin E goes as exp(Ng(cr)) for N large; thus for large N almost all EC’s have the energy/spin E, , where g(E) attains its maximum g,,, at 6 = E1, and the relative standard deviation of energy is

u EC = N- I’* 1 g”(cl)l -I’*,

where I g”(el)l = I d*g(e)/d&* IEZE,. Although the characteristics of EC valleys (e.g., the distributions of local fields or relaxational eigenvalues) do vary with their energies, this variation is unimportant because the range of possible energies is quite small (in the case of vector spins) [ 151. (This is why calculations [ 1, 3, 9-141 based on EC’s produced by the relatively crude “Walker-Walstedt” algorithm give meaningful results.) Thus it is plausible that the configuration-space volume of the basin of attraction of an EC-and hence the probability that the Walker-Walstedt algorithm will reach it from random starting configurations-should be almost independent of the energy E of the EC. The resulting sample of EC’s is distributed in energy according to the true density of minima Nmin(&), without any bias toward low

energies, so the numerical averages over this sample are “white” in the sense of De Dominicis et al. [ 161. In particular, we find those EC’s which have energy e g E~ where N,,+(E) is sharply peaked, not the rare near-ground states of energy E z c0 (where g(cJ = 0 defines the ground state energy E,,). Finding near-ground states should be very interesting but requires some sort of annealing process which is beyond the scope of this paper.

Using the model of identical, independent clusters from Section VII of the preceding paper [6], one can use the measured oZc to estimate the actual ground state


energy and the “excitation energy” per cluster, E,. Copying formulas from Eqs. (7.6), (7.4), and (7.9) of Ref. [6],

112 N, ‘E, ,

g max = g(c,)= (log 2)N,‘,

Also, substituting (2.7~) into (2.6) gives

CJ EC = (nN,/4N)“’ (cl - co).

(2.7b)

(2.8)

Accepting the estimate [ 151 g,,, s 0.008, the cluster size is N, z 86 (from (2.7b)), the number of clusters per sample is N/NC z 20, and we have E, - s0 z 4.5 X lop3 from (2.8) and E, = (7c/2)“’ N,(E, - EJ z 0.5 from (2.7a). This agrees with the independent estimate of E, from defect-insertion experiments.

For one of the realizations I have calculated the distribution of exchange fields hi. and Fig. 2 is its histogram. This should be compared with the analogous plot for the RKKY spin glass, Fig. 7 of Ref. [9]. This histogram’s bimodal appearance is an artifact of statistical fluctuations. The smallest and largest fields are about 0.55 and 9.35; the average is 3.71 units with a variance (1.4 units)‘. However, the field on any one site varies much less between different EC’s; its variance, (hf), - (hi):, is about (0.14 units)’ except for the high and low tails of the distribution, where the variance is much smaller. This is explained by the fact that locally, the spins are rather rigid and one EC just looks like a rotated version of another.

“, FIG. 2. Histogram of distribution of the exchange fields (hi) in four equilibrium configurations of

one realization of couplings in an N = 12 x 12 x 12 nearest-neighbor Gaussian spin glass (solid curve). Note that a height of 100 sites per bin corresponds to (0.289) N sites per unit of hi. The dashed curve shows the distribution for four configurations of randomly oriented spins.

332 CHRISTOPHER L.HENLEY

III. COMPARISON PROCEDURE

Once we have generated two EC’s of a given realization, say, {s’i”} and {$“}, the next step is to compare them and extract information on their relationship. This procedure has two parts: (a) to extract the local (relative) rotation matrices, and (b) given a configuration of rotation matrices, to define disclinations. After describing the algorithm I used, I will discuss the validity of its results.

A. Rotation Matrix Configurations

Here I explain how I implemented the reduction of two spin configurations to a configuration of relative rotation matrices, reviewing the discussion in Section II of the preceding paper [6] of the definition of the “order parameter” matrix Q(Z) and the singular value decomposition by which it is projected to a rotation matrix R(Z). (Note: I will use “rotation matrix” as a synonym for “orthogonal matrix.” For brevity I use “sign of R” to mean “sign of det(R).” “Proper rotation” specifies a rotation of positive sign.)

I first computed the overall “order parameter” matrix

(3.1)

and the locally averaged “order parameter”

here ?i,i is the lattice vector ,?- Zj, reduced module the periodic boundary conditions so as to minimize ]yiji. I chose the weighting function ~(6~) to be l/8, l/16, l/32, and l/64 for ri = 0, 1, 2, and 3, respectively, for a total of 27 spins contributing. This means that the number of spins which contribute with large weights was effectively about 10, a number which was found to be optimal for both the nearest-neighbor spin glasses reported on here and also (in trials) for RKKY coupled spin glasses. (By “optimal” I mean this averaging volume was large enough to include spins pointing in all possible directions, but small enough that the spins included are tied together fairly rigidly.)

The overall order parameter was decomposed by the singular value decomposition

0 = R,DRT,, (3.3)

where R, and R, are 3 x 3 (O(3)) rotation matrices and D is a diagonal matrix whose elements {dy, y = 1,2,3 } are the singular values of 0. This decomposition is unique apart from choosing the order and signs of the d;s (assuming their magnitudes are nondegenerate). Then [6] the overall rotation matrix which best relates the configurations is given by

R=R,R;. (3.4)


Equation (3.4) does not yet fully determine R, since its value depends on the choice of signs of the d;s. The right choice is

as I now show.

(35)

The measure of how nearly the overall rotation matrix R makes the two configurations coincide is the projection

(3.6)

(where equality means a perfect rigid rotation). As a function of R, Eq. (3.6) has two local minima in the manifold O(3): one proper rotation, one improper rotation. For R defined by (3.4), Eq. (3.6) becomes

P=xd, (3.7)

and the two minima are seen to correspnd to the choice (3.5). Normally I chose the h corresponding to d, > 0, which gives the global maximum of P. For every EC generated from random configurations, there is another, equally probable EC related to it by a simple reflection, so there was no reason to prefer one sign of R. However, in the comparisons reported in Section V, one EC was produced from another by spatially varying, but proper rotations, followed by relaxation; in this case the most meaningful choice was to take R to be a proper rotation, which occasionally forced choosing d, < 0.

Next, I decomposed the locally averaged order parameters

and extracted

iTi = Ri, R& (3.9)

for each site, in exact analogy to Eqs. (3.3t(3.5). In this case I always chose R, to have the same sign as R; usually this forced (dJi < 0 on a few sites. I will call these sites of “wrong sign.” In these places, the best-fitting rotation would have had a sign opposite to the sign predominant in the sample; that choice would have introduced sharp singularities in the configuration of rotation matrices where the sign changed, which (usually) did not reflect any singularities of the relation of the spins configurations. Note that a domain in which spins have been reflected (a defect discussed in Ref. [6, Sect. III ff), analyzed by the above algorithm, became a domain of wrong sign. Thus the local values (d,)i are physically interesting, as ‘well as being diagnostics of regions of “wrong sign.” The {(d3)i} were the only tool I had for studying the reflection domains and walls modeled in Section IV of the preceding paper [6] and further discussed below.


I eliminated effects of an overall uniform rotation as follows: imagine uniformly rotating configuration letting

s!l)! = ,p I (3.10)

for each i. Then comparing configurations 1’ and 2 must give

ii’ = 1,

Q’ = R,DRT,, (3.11)

R; = ET&

(1 = identity matrix). Note that R,f must be a proper rotation, since I chose sign Ri = sign R. I used these rotated order parameters for all further analysis; therefore I will henceforth omit the primes (and the overscores on Ri, since that is always locally averaged).

I also computed comparisons of configurations with themselves, generating Q(“) and {Qj”“‘}. Th is might seem uninteresting, since all the R~“‘)‘s will automatically be the identity, all the (d,):s will be positive, and PI@,“‘) = tr Qi““) = 1. However, each Qjwp’ matrix contains potentially useful information: its traceless part gives the I= 2 spherical harmonics of the distribution of spin directions in the averaging volume around site i. The eigenvalues of &, the ((d,)i}, are the principal moments of this distribution, and the eigenvectors (rows of RLi = RRi) are the principal axes. A value (d3)i z 0 indicates that the spin configuration near spin i is XY-like, i.e., all spins are confined to a plane.

It is of interest to calculate the expectations of the singular values, both the overall ones (6,) and the local ones {+}, assuming cases of uncorrelated or correlated random spins. This has been worked out in Appendix B.

It was convenient to represent the (proper) SO(3) rotations Ri by four-component unit vectors uli = (cos $9, sin f@, where Ri is a rotation by an angle 19 about the axis 8, as described in Ref. [6].

It is useful to have a pictorial representation of the configurations of rotation matrices, similar to that used by Grzonka and Moore [3] for their simulation of an XY spin glass. With the XY spins represented by angles Bi, the relative rotations are represented by the angles Ad, = 01” - 0;” (no local averaging necessary), which can be plotted as arrows oriented at an angle AO, from a fixed direction (compare Fig. 1 of Ref. [3] and Figs. 3-6 of Ref. [6]). This is impossible for Heisenberg spins since each matrix corresponds to three degrees of freedom (three Euler angles). I chose to plot only one of these, the “projected angle” #i representing the rotation of Ri about a specified axis a^. When comparing independently generated minima, the axis a^ was chosen to be the eigenvector of Q corresponding to the largest singular value; the two contigutations are best matched for spins pointed in the directions *a”, so one gets the most information by looking at the spin components in the other two directions. Given a^, I defined the projected angles as follows: imagine one projected the spins of configurations 1’ (see (3.10)) and 2 into the two-dimensional subspace perpendicular


(in spin space) to a^, thereby making configurations of variable-length XY spins; one could then substitute the XY spins into (3.1) to define a 2 X 2 Qi;,,. I actually implemented this by directly projecting the 3 x 3 Qi normal to a^. Then Qiixy was reduced by the singular value decomposition to give a 2 X 2 proper rotation matrix, and Qi was defined as the corresponding rotation angle. The projected angles could then be represented by vectors, as in Figs. 3, 7, 9, and 10; one should keep in mind that they do not represent directions of spins (which look quite random) but differences between one configuration and another.

B. Disclinations

The last stage of processing was to identify the disclination lines. To make an operational definition of a disclination given an array of rotation matrices, it was convenient to start with the representation of the rotation matrices R(Z) by unit 4- vectors u^(x’). If R(x’) rotates through 360” as we traverse a loop in space, then u^(-?) rotates through 180”. Thus, somewhere along the loop there must be a jump u^ -+ -u”, which is not a singularity in R(Z) since +u” and -u^ both represent the same rotation. Most precisely, there is an odd (even) number of such jumps if the loop encloses an odd (even) number of disclination lines. If we had a continuous field of rotation matrices, this would allow an unambiguous identification of disclinations. To implement this idea on a lattice, I took each nearest-neighbor bond connecting sites i and j, and if uli n Gj < 0, I said there was a jump along that bond. When one or three of the four bonds of a plaquette had jumps, I said there was a “nominal disclination line” piercing that plaquette. (I call them “nominal” since disclinations can be defined by this rule for two arbitrary spin configurations, thus the results do not necessarily correspond to anything meaningful.) Thus the nominal disclinations are composed of segments connecting the centers of unit cells; they are walks on the dual lattice. One easily checks that nominal disclinations obey a topological conservation rule: each cell must have an even number of disclinations piercing its faces, and so the disclinations connect up into closed loops (or lines which run from a point on one face to the identified point on the opposite face, which are another kind of loop).

Alternatively, one could have defined nominal disclinations as the apparent vortices in the configuration of “projected angles” (treated as XY spins). It is reassuring that the disclinations are in about the same locations by either definition (see Fig. 7).

There was always noise in the reduction to disclinations. This was due to the discreteness of the lattice and to the disordered nature of spin glass EC’s, which have some spins which are very rigid and others which are very sensitive to perturbations [9]. For instance, around a plaquette pierced by an ideal disclination, the rotation would change by 90” on each bond, giving Gi . z;j E 0.707, which is comfortably positive. In reality one bond often had about 180’ of the rotation, and this could spuriously change the sign of uli . Gj on that bond: the symptom of this was a spurious disclination loop of length 4 encircling that bond on all sides. Four such “noise” loops are visible in Fig. 6.


A procedure of “disclination editing” was used to remove the sharpest kinks in the disclination lines and small noise loops. If three sides of a dual plaquette had disclination line, it was moved instead to the other side; if four sides had disclination, this loop was eliminated. This made it easier to identify genuine disclinations by visual inspection of the outputs; however, the nominal disclinations shown in the figures are unedited. I computed the fraction of the links of this dual lattice that actually carried disclination lines (after editing). This quantity, which I call the “disclination fraction,” was often a good measure of whether an inserted defect had remained or gone away.

Finally, the most useful information extracted from the Qi’s from comparison of different EC’s was printed out layer-by-layer, representing the projected angles an the singular values dji (which show the regions of “wrong sign”) by one-letter codes on each site, and representing the disclinations by lines. This output was analyzed by hand.

C. Questions of Validity

It should be emphasized that the algorithm described above can take anq’ two spin configurations, including random configurations, and produce local rotation matrices and nominal disclination lines. Therefore, one must test whether the results are meaningful. The main test of the rotation matrices is to look at the optimal projection

‘Ypt = Cy lCdJi I a each site i, which has its maximum possible value of 1 if the t

rotation is completely rigid in the neighborhood of site i. For the scheme of weights w(ci) which I used, Ppp’ averaged 0.87 for comparisons of the independent minima reported in Section IV (because of “wrong-sign” sites, the actual projection Pi = C, (d,), averaged 0.80). This agrees with the predicted projections which can be calculated given the correlation functions defined in the next section (see Appendix B). For comparisons of configurations of randomlpv oriented spins, Ppp’ averages only 0.28. This too agrees with the calculation in Appendix B.

To test whether any information has been lost in going from raw spins to rotation matrices, one trial of the following experiment was made: (i) two EC’s were generated, (ii) a set of RI)s was generated in the usual way, so each Ri maximizes the projection of (Riq”} on (?j2’} for sitesj in the neighborhood of i, and (iii) I defined a third configuration {s’j3’} = (Ri$“} and relaxed it until it was an EC. It relaxed rapidly to (s’i”}. (To explain the rapidity, note that all the configuration’s deviations from equilibrium come from the localized eigenvectors of the Hessian matrix-i.e., relaxation modes-rather than the long-wavelength modes. As discussed in Appendix A, the localized modes have large eigenvalues and hence relax quickly while the extended modes are slow.) This test confirms the validity of defining locally averaged Ri’s: that is, it is possible to find, and to use for averaging, a length scale which is large enough to include spins pointing in all directions but which is smaller than the correlation length rQT of the rotations. The slowly varying Rts suffice to uniquely represent the EC’s (relative to some reference configuration which can be arbitrarily chosen from the set of EC’s).


The main situation in which rotation matrices and nominal disclinations were usually meaningless was in a region of “wrong sign,” a part of the sample where the true local rotation matrix disagrees in sign with most of the sample. But my algorithm forced the sign to be the same everywhere; the result is that the projection in the “wrong-spin” regions, P = d, + d, - 1 d, 1, was poor (unless d, is near zero, the significance of which is discussed in later sections). When my algorithm analyzed such regions to produce nominal disclination lines, the result was typically a spaghetti of spurious lines (a cross section of which is indicated by the stars in the shaded parts of Fig. 3). A spurious line typically looked like a random walk and often went long distances, crossing the “wrong-sign” region several times, before reconnecting to itself. In contrast, lines corresponding to a true defect were often simple closed loops.

There was little hope of identifying disclinations relating a pair of independenrlv generated EC’s of a given realization by inspection of my algorithm’s output, since in this case true rotation matrices typically had equal volumes in domains of each sign. Some information can be derived for this case from the correlation functions, as I discuss next.

IV. CORRELATION FUNCTIONS

If a defect picture were good, one should be able to compare any two independently generated EC’s and locate well-defined defects. This failed for the N = 12’ samples because it turns out that <or, which is the typical defect separation, is only three times the nearest-neighbor distance, so one cannot cleanly separate defects from each other and a large frection of the sample is in “cores.” In particular, half the spins are in “wrong-sign” patches, and the patches of a given sign have have radii 4QT ; as elaborated in the preceding section, it is pointless to look for disclinations and twists under these circumstances. I show such a comparison in Fig. 3. Of course, the regions of “right” and “wrong” sign exchange places when we change the overall sign used for the comparison, but the rotation matrices look completely different.

Therefore, I studied the independently generated EC’s by calculating four different correlation functions, C,, C,,,,, Ccr and CSig,, from the EC’s of each realization. The first two of these refer to the spins within a given configuration, and the second two are “relative” correlation functions which refer to a comparison of two EC’s. I will now give the numerical definitions and results for the correlation functions (see preceding paper [6] for a fuller discussion); in particular, from the range of the correlations of Q, I will verify what was hard to see in a raw comparison: that the spins are locked together over a relatively large volume.

A. Correlations within EC’s

I generated a total of n = 4 independent EC’s for each of four different realizations of a 123-spin sample. For each realization, for each of its n EC’s, I defined

(4.1)


FIG. 3. The projected angles di relating two independently generated EC’s of the same realization of a I2 X 12 X 12 sample of spin glass as in Fig. 2. Only one of the 12 layers is shown. A difference in angle between the EC’s of O” in the neighborhood of a site is represented by an arrow at that site pointing to the right; see text for definition of how di is derived from the local relative rotation matrices Rj. In the comparison algorithm (described in Section III), the signs of the Rts were constrained to be everywhere (a) positive (algorithm assumes EC’s are related by a paper rotation), and (b) negative (improper rotation). The sites where the sign assumed is “wrong,” i.e., opposite from the best choice of sign af that site, are shaded. Stars indicate points where the layer is pierced by a nominal disclination line, as calculated by the algorithm of Section III.


(Here and henceforth, irij is the lattice vector Zj - Zi, reduced modulo the periodic boundary conditions to get the minimum ] c7 I.) This measures the correlation in spin orientation: to what extent q is parallel or antiparallel to K, as opposed to being perpendicular. A value C,(7) = 1 means we have an Ising-like EC.

Furthermore, I compared each EC with itself, calculating the locally averaged (and in this case, symmetric) matrix qi“*“’ at each site i as described in Section II. Then I let Fiji) be the eigenvector of Zeast eigenvalue of (Qj”“‘); that is, if the spins of EC number p in the neighborhood of site i tend to lie in a particular plane, then Gi“) is the axis normal to that plane. I defined

(The dot product in (4.2) is squared since there is no way to choose which of +si is the normal axis.) Thus Caxis measures, more sharply than C,, the tendency for the EC to look like an EC of XY spins. Unlike C,, Caxis is smeared out by-roughly speaking, twice convoluted with-the averaging function used to define @‘“‘. (This local averaging is unavoidable in the case of Caxis since we need more than one spin to determine a plane.)

For each realization, I calculated C,(J) = n-’ 2, C?)(F), an average over the M different EC’s, and similarly C,,,,(7).

B. Correlations between EC’s

The most important correlation function for a spin glass is the “order parameter” correlation function, defined for each pair of EC’s (u, V) by

When this is averaged over all pairs (,u, V) it is the Edwards-Anderson correlation function appropriate to the weighted ensemble of configurations defined by the Walker-Walstedt relaxation from random initial conditions (see preceding paper [6] and references therein). This correlation function measures correlations of the relative rotation matrix, our “order parameter”; if C,(yij) is near 1, it means spins i andj are rigidly locked together so that .?,. . c is the same in every EC. However, (4.3) also includes contributions from the same (longitudinal) fluctuations as C,; projecting these out, I defined a transverse “order parameter” correlation Cb”;” by

that is,

Cb”“’ = 3 (@’ + Cl,“’ + C@‘), (4.4)

which I calculated for each pair (,u, v).


Finally, I compared each pair (u, V) and decomposed ~~$’ by the singular value decomposition, as described in Section II, extracting the relative sign aj”) = sign(det(qj’“‘)) = sign((d,),) = i 1. I then defined

The correlation function Csign tells us how far away we can go from a point and describe the variations in our “order parameter” using only proper rotations; in other words, how far we can go before crossing a “reflection wall” defect [6]. This correlation function is also smeared like CaXi,.

For each realization, I calculated Co,(?) = [n(n - 1)/2] -’ C,+,, Ct$‘(7), and similarly Csien, an average over the n(n - 1)/2 distinct pairs of EC’s.

From their definitions, Eqs. (4.1) and (4.3) one finds [ 6 ] sum rules relating C, and Co to the overall Q matrices,

xs E 2 c(sLL’(J) = N (; @lcb”‘)* - “) = N; (d:,” - f)*, (4.7) i

xo = x cw”(y’) = Nx (@;‘)’ = Nx (,:,“‘)*, i a4 ‘/

(4.8)

which are exactly true for each EC (for (4.7)) or pair of EC’s (for (4.8)); they thus served as numerical checks. Numerically. the average values were

C. Results

xs = 5, x0 z 90, xQT = 3x0 - 2xs s 260. (4.9)

All four correlation functions were calculated for 0 < 1 r, 1 < 5 (here a is the component index (r = 1, 2, 3; in these 123-spin samples any larger displacements would be ambiguous, due to periodic boundary conditions); the results are plotted in Fig. 4. In summing the correlation functions, all points equivalent under permutations and reflections of the lattice vectors (e.g., [ 1, 2, 31 and [ 3, 2, -11) were grouped together and represented as a single point in the figures; however, inequivalent displacements of equal length (e.g., [2, 2, 1 ] and [3, 0, 0]) were accumulated and plotted separately. This was done because the lattice makes the correlations anisotropic: they are somewhat closer to being a function of the “Manhattan metric” distance C, ] ra 1 than of the “Pythagorean” distance (C, ] Y, ]2)1’2. which is used in Fig. 4. This explains the apparent scatter in C,(r) in Fig. 4a.

One sees that there is no well-defined exponential decay of the correlation functions, but one can estimate nominal “correlation lengths” <, = 2 [log(C,(r = l)/C,(r = 3))] ‘, where “x” = “s,” “QT,” “axis,” “sign.” For each realization, I averaged the correlation functions C,(q and evaluated (4.10) from these averages I then averaged the resulting &‘s over the four realizations; the results are presented in Table I. Because of the local averaging needed to generate the Qp s, the values of taxi, and &,, are always spuriously increased.


lOO* I (01

0.01 I 0 3 6

1.01

COT

0.11

0 3 6

10

C sqn

01

t

o-

OOI- , 0 5

. .

Frc. 4. Semilogarithmic plot of the four correlation functions (a) C,(r), (b) Caris( (c) COT(r). and (d) Csign(r), defined in the text, as a function of the offset distance r. The correlation functions are averaged over the four EC’s generated independently for the realization used in Fig. 2 and Fig. 3. This realization has the most typical correlation functions in that the corresponding ad hoc correlation lengths 5,, taxi,, loT, and L,, are closest to the average values reported in Table I. The apparent scatter in C,(r) is due to the anisotropy of the correlations on a lattice.

TABLE I

Averaged “Correlation Lengths” for Four Correlation Functions

Correlation length In EC’s Random spins

0.6 0 0.8 0.5

t;,T 3.4 0 L" 0.9 0.5

Note. Values of the nominal “correlation lengths” defined by (4.10). First column: values from EC’s averaged over four realizations (four EC’s per realization). Second column: values from six configurations of randomly oriented spins, analyzed as if they belonged to the same realization. Note the spuriously nonzero values of &, and &,,, due to the local averaging (see text).


Some realizations gave longer correlations than others; the realization used for Fig. 4 is “most typical” in that the nominal &‘s are closest to the averages over the four realizations reported in Table I. The correlation lengths vary within a given realization, depending on the EC’s, probably because of variations in the degree of relaxation. Indeed, as one would expect, the inter-EC correlation lengths roT and rsign increase with longer relaxation times, but this increase is already saturating at the time when the minimization algorithm halts (r, and taxi, saturate much sooner). A tendency I observed for the longer-range correlations to decrease slightly below zero at long times seems to be an artifact of roundoff errors.

We see from Table I that C, and Caxis, the spin correlations within the same EC, have a range of the order of one nearest-neighbor distance; this agrees with Walker and Walstedt’s result [9] for the RKKY spin glass (which used a number of definitions of spin correlation volume, none of them identical to mine). On the other hand, Co, and Csign have a longer range; in particular lQT = 3 to 4, depending on the realization and on how well relaxed the EC’s are. The corresponding correlation volume, which should be approximately proportional to $,, is xQT z 260 spins (see (4.9)). One would expect rsign to be about the same as xoT and it is unclear to me why it is, in fact, shorter.

While comparing independently generated EC’s, I also evaluated the projections P(p, v) = c, dy. I found P g 0.38 on average (this should scale as N-l”, but with a large prefactor!). One could make an estimate of the “patch size” from this quantity, but it is really the same as the estimate from xOT, owing to the close relation of P and xOT which can be seen from (4.8) and Appendix B. For configurations of purely random spins, PE 0.06. These values agree with the values following from the calculation in Appendix B of (d,), given x0 (see Eq. (B.27) with (B.30) and (B.23)).

V. INSERTION OF DEFECTS AND TEXTURES

The main purpose of the work reported in this section is to prove the existence of the two topological objects associated with proper rotations, the disclination line defect and the “twist” texture, which were discussed in Ref. [7] and in Section III of the preceding paper [6]. My method consists of insertion, relaxation, and comparison. I start with one of the EC’s generated from random initial conditions (most of these were also used for computing correlation functions; see Section IV, above). Also, I make a configuration of rotation matrices which cleanly represents (a) a 360” twist, or (b) a pair of disclination lines running from one side of the sample to the other. Then the rotation matrices are applied to the EC to produce a new configuration which indubitably has that topological object (and no other) relative to the starting EC, but which is not an EC; this is the insertion procedure. This configuration is relaxed to an EC, at which point we compare the two EC’s to see if the defect is still identifiably there.

My insert-relax-and-compare procedure is reminiscent of Reed’s computer experiment [ 11, done using the same model as reported here (Heisenberg spins,


Gaussian random couplings, simple cubic lattice). He performed a rigid rotation of a block of spins within the sample, leaving the bulk outside that block undisturbed, relaxed the entire sample, and asked whether it returned to the old configuration or to a new one (and, in the latter case, what the energy difference was). My procedure differs in that (i) I insert a topofogically stable object, so that in principle there is a barrier preventing it from relaxing back to the starting state. (ii) Reed compared the two configurations by looking at the average angle that individual spins rotated between the two configurations, while I have a more sophisticated comparison algorithm.

A related procedure of “heating” an EC for 100 Monte Carlo steps, followed by relaxation to a new EC and comparison, was used for XY spins in d = 2 [ 171.

An insert-and-relax procedure was also used in the simulation of Chaudhari et al. [S]: they inserted dislocations into a realization of a model of a metallic glass, and then relaxed the atomic positions. They tested for whether the dislocation stayed by looking at the atomic displacements from the starting configuration to the final relaxed one, a comparative criterion analogous to those I used for the spin glass. However, as an independent test they also computed the Airy stress function in the relaxed sample to see whether a dislocation was present; this is an absolute criterion which does not require information from the starting configuration. Analogously, in the case of the Heisenberg spin glass, the quantity

t: Jij[(Cj)j,, (< X q>,l (5.1) (iA

summed over the entire sample is proportional to the average angle gradient (aO,/ax,), and thus can give an absolute measure of twist without a need for comparisons. The quantity (5.1) was not calculated in the experiments reported here.

In the rest of this section I will discuss the procedures and the results (qualitative and quantitative) of these experiments, first with the twists and second with the disclination pairs. Before concluding, I discuss the importance of the ubiquitous “reflected regions” which I believe may be an especially important kind of defect (note that I did not try inserting them!).

A. Insertion of Twists

Starting from one EC (call this EC”‘), a “bare” twist was applied to give a configuration T(EC”‘). Within each of the 12 layers perpendicular to the twist direction the spins were rotated uniformly about the twist axis, by an angle which increased by 27r/l2 from each layer to the next. At this point a comparison of EC”’ and T(EC”‘) would show a 360” twist but no disclination lines or other singular defects. Then T(EC”‘) was relaxed by the Walker-Walstedt algorithm to a new equilibrium Ecu), using the same convergence criteria (described in Section II) as for EC”’ (this relaxation converged somewhat faster than that of EC”’ since T(EC”‘) was already well relaxed at short scales). Note that, up to this point, my procedure was the same as the one that Walstedt [lo] used to evaluate the spin-stiffness


constant, except that he always applied small-angle twists whereas I always apply a 360” twist.

Next, EC’” was compared with the bare-twist configuration T(EC”‘) to produce local rotation matrices, values of (d3)i, and projected angles $i as described at the end of Section III. TO understand why it is valid to use T(EC”‘) when in principle we should compare with EC”‘, we should consider the following composition rule: If we have three spin configurations {sj”)}, p = 1, 2, 3, and the pairs (1, 2) and (2, 3) are perfectly related by slowly varying rotations R(‘-*)(2) and R(2q3’(.?), then the pair (1, 3) is also related by a slowly varying rotation R”‘“3x() = R”**‘(.?) R’**“(.?). This rule is valid in the continuum limit, in which the averaging volume used in defining R(Z) becomes arbitrarily small relative to the characteristic length over which R(.?) varies. (This continuum limit cannot be realized by varying the parameters of my Hamiltonian (1.1); it is an idealization which was used, for instance, in Section III of [6].) In particular, R”,“(?) = R’t.To’(X’) R’ToYo’(2), where superscripts “0,” “TO,” and “t” refer to the configurations EC”‘, T(EC”‘), and Ecu’, respectively. Since R’ToVo’(Z) has no singular defects (disclinations or reflections), and only the inserted twist, it follows that R’t,To’ (x’) has the same singular defects as R”,“‘(x’), differing from it only by that twist. In practice we are far from the continuum limit and, in the comparison R @*O), the rather large uniformly angle gradient of 30” per layer tends to spuriously enlarge the disclination loop defects that appear during the course of the relaxation; therefore I used R’t.To’ instead, in order to subtract out that gradient.

The projected angles #i were taken about the same axis a^ as was originally used for the twist. Presumably many of the changes during relaxation are further rotations about 8, so the projected angles usually show the most significant degree of freedom of the EC. However, it is conceivable that after relaxation EC’” might be a clean twist of the original EC, but about an axis orthogonal to a”; in such a case the calculated 4;s would be garbage.

An additional comparison was made of EC”’ and EC”’ in which one G matrix was generated for each layer with each of the 144 spins in the layer contributing equally, and then the Q’s were reduced as usual. The resulting “layer-projected angles” 4,,,,, 3 which appear in the insets to Figs. 5 and 6, are useful diagnostics of how well the inserted twist “stayed.”

The entire insert-relax-and-compare procedure was repeated on a total of 13 EC’s of the 5 realizations that were used. To minimize distortions from having such a small number of realizations, I used each starting EC only once, and furthermore I used different directions in real space for the twist gradients for different EC’s of the same realization. The output (showing (d3)j and #i on each site i and the nominal disclinations) was analyzed by hand.

To understand the results qualitatively, it helps to start with smaller samples (say, 63 spins), which have only a few distinct EC’s. In this case it is most common for the twist ro relax away completely and EC”’ is EC”‘, perhaps with some local defects. It is instrcutive to picture the stages of this process. At first, a disclination loop is nucleated somewhere such that, along paths which thread the loop, the twist is zero, and so the gradient energy is decreased. Clearly there is a force on the loop to make


it expand, which it will do until, because of the periodic boundary conditions, one side of it touches the other side and annihilates. We now have a pair of disclination lines where the (smooth) spin rotations in between favor a continued expansion, i.e., the lines attract through the identified sides until they touch and annihilate along part of their length, producing a disclination loop where now the twist is 360” on$ along paths which thread this loop; this loop finally shrinks to zero.

It is always possible that the disclination loops will get pinned due to the microscopic randomness of the spin glass, and the annihilation process will get hung up at some stage. As one goes to larger systems, in which the 360” twist is spread over more layers so the gradients are smaller, this outcome becomes increasingly likely. The end result EC”’ may have a dilute spinkling of disclination loops (Fig. 5 has two of them) or it may have a pair of disclination lines from end to end (Fig. 6).

There are clearly domains of “wrong” (reflected) sign, some of them associated with disclination line, others not. Around the edges of these domain the EC’s are usually not related by a rigid rotation; within them there may be a rigid rotation which relates them, but the comparison algorithm will not produce it, since the algorithm forces all rotation matrices to have the “right” sign. On the other hand, the algorithm can interpret any configuration in terms of disclinations; consequently, strongly reflected regions have a “spaghetti” of spurious nominal disclination lines. Reflection walls are topological defects in their own right; I elaborate and speculate more on them in part C of this section (below).

One should question whether the nominal disclination loops which appear during relaxation of a twist are real disclination loops. They are always associated with regions of wrong” sign (reflections); perhaps the loops are just bits of the random “spaghetti” mentioned in the preceding paragraph. There are two pieces of evidence that they are good disclination loops:

(i) They tend to be oreinted to lie in a plane normal to the direction in which the twist was applied. For example, the loops in Fig. 5 show large areas when projected in the 2 direction, the direction of the twist; a projection in the 3 or I* direction would show much meandering but a small enclosed area. (Most precisely, the oriented area A defined by I,,,, r’X dr’= Aii has a unit vector n” almost parallel to *.t)

(ii) The projected angle fields show that there is no twist along paths threading the loop.

Although, then, I believe the disclination lines are not mere artifacts, I also believe the associated reflections are genuine, so that the defect is best interpreted as a combination of the two, as discussed in part C of this section.

To determine whether the twist “stayed,” I used several numerical criteria (as well as visual inspection of the outputs representing the rotation matrices and the nominal disclinations). In the three runs in which it did “stay”:

(i) The change per layer LQ&,,,~~ of the layer-projected angles was at most 75’-120’ in all layers.


(ii) The projection P of EC”’ on T(EC”‘) was at least 0.6 and was greater than the projection of Ecu’ on EC”‘.

(iii) In the comparison of EC”’ and T(EC”‘), less than 1.5 % of the plaquettes had disclinations piercing them.

To understand why I say a twist “stayed” even when A@,,,,, = 120’ in one layer, note that (1) the bare d4,ayer is 30”, already rather large; after relaxation Ad,,,,, s 15”-60” because of the disorder. (2) In layers where a small loop has been nucleated, 4 ,ayer can have a sudden jump even though the Rts vary smoothly (a smaller version of the jump in Fig. 6, inset, due to a disclination pair).

To gauge the significance of P > 0.6, and for future references, it is of interest to consider the projection of the bare-twisted T(EC”‘) on EC”‘. If the twist is about axis a^ in spin space, then

e,, = 3 ua a, + O(W 1’2). (5.2)

Consequently P(EC”‘, T(EC”‘)) g 3. For comparison, the average projections after relaxation are P(ECct’, EC”’ ) G 0.46, P(EC”‘, T(EC”‘)) z 0.55; thus, in any case,

/-,-l, ~&lle*

3600

c _,_

hyer ,'

180' ,,'

El

,' ,' ,-

OO 6 12 x

FIG. 5. The disclination lines relating a twisted-and-relaxed configuration, EC”‘, to the untwisted configuration EC”‘, are shown (solid lines) in orthographic projection. (Most precisely, EC”’ is compared with the “bare-twisted” configuration T(EC”‘) as explained in the text.) The twist was a total of 360° over 12 layers applied in the I-direction. In this case, the twist “stayed”: EC”’ differs topologically from T(EC”‘) only by two disclination loops. The loops show a large projected area in the T-direction (the projections are shown hatched and their areas in lattice units are indicated). There is no twist along the dashed lines which thread the disclination loop, as was checked by inspection of the projected angles along the dashed lines. The inset shows the layer-projected angles ),,,.r derived from averages over each layer (solid curve), which shows only minor changes from the ),ayer’~ of the bare twist (dashed curve).


these configurations are more closely related than the independently generated minima (Section IV).

Figure 5 shows the disclinations in a typical one of the three runs in which the twist “stayed.” There were also three runs in which the twist “decayed” in that EC”’ was more like EC”’ than T(EC”‘). In the remaining seven runs, the twist decayed partially, giving rise to disclination pairs in three of these runs. These are in fact better specimens of disclination pairs than most of the configurations produced by inserting and relaxing a disclination pair (see B, below). One of these comparisons is plotted in Fig. 6, which shows the disclination lines, and in Fig. 7, which shows the projected angle fields in one layer where they clearly show the vortex-like pattern of rotations associated with the disclination pair.

I also studied the energy differences between the initial and final states. First, note that the bare spin-stiffness constant P?’ is defined in terms of the energy difference when we apply a uniform twist (with an infinitesimal angle gradient) and do not relax the spins [ 181; one can easily derive that pp’ = (2/9) /eOI with nearest-neighbor interactions on a cubic lattice (the length unit is a lattice constant), i.e., pr’ = 0.41 units for the couplings used here. Thus T(EC”‘) should be higher in energy than EC”’ by 2n*p,/L* = 0.056 units per spin or a total of 97 units for the sample, or 95

FIG. 6. Disclination lines relating a pair (EC , ‘O’ EC’“) from a different realization than in Fig. 5. In this case the twist “decayed” into a pair of disclination lines. The small squares indicate points identified according to the periodic boundary conditions. The circled segments indicate the layer z = 6 which is represented in Fig. 7. The dashed segments are spurious disclination loops due to noise or to overturning of a single spin. The inset shows the layer-projected angles as in Fig. 5. (The disclination loops sit roughly at x = 4 and x = 8, corresponding to the flat parts of the curve in the inset: between the loops

4 ,ayer has a jump of 19Y.)


FIG. 7. The projected angles 4i relating the sme pair of EC’s as in Fig. 6 are shown in the layer z = 6 using the same conventions as in Fig. 3. The stars show where the two disclination lines pierce the layer: in this layer the ii’s have a clear vortex-like configuration near the disclinations. Note that these angles actually represent the difference between EC”’ and the bare-twisted configuration T(EC”“). as

explained in the text. so that the apparent lack of twist in the 1 direction actually means that the 9 twist stayed. The apparent recerse twist of 360” along a path passing to the right between the disclinations means that there is zero twist of the relaxed state relative to the initial one; i.e.. in between the disclinations the inserted twist has “gone away.”

units when we take into account the term quartic in the gradient (which is a large 27412 per layer); the observed difference averaged 95 units with a scatter of +5 units.

Now, if no defects were nucleated as the twist was relaxed, the energy difference between EC”’ and the relaxed configurations EC”’ would be E”’ -E”’ = 2n2p,N/L2, where ps is the (true) spin-stiffness constant, as in Walstedt’s calculation [lo] in which the gradients were small. Computer experiments give ps = 0.12 (Ref. ( 111) and ps = 0.17 (inferred from Fig. 1 of Banavar and Cieplak [ 12]), so the total expected energy difference E’” -E”’ is about 30 units. In fact the energy difference was very small: for the eight cases in which both EC”’ and Ecu’ were nominally “converged,” Pt’ -E(O) = -0.1 f 3.3 units, ]E”’ -Z?” 1 = 2.7 i 1.6 units (the errors here are the r.m.s. deviations). If instead I take the three cases in which the twist “stayed” (in two of these cases EC”’ was not nominally “converged”), then p _ E’O’ = IE’” -E”‘/ = 1.3 f 0.5. It is not surprising that the energy has a nonquadratic dependence on the amplitude of the twist gradient, and that in spin glasses this contribution is already dominant for small gradients. My gradients are already too large to probe the crossover from quadratic dependence, which might further clarify the question of whether the spin-stiffness constant is actually zero [3]. However, it is interesting how, in this large-gradient case, defect nucleation could relax almost all of the gradient energy while still leaving most of the twist intact.


B. Insertion of Disclination Pairs

A second series of experiments was carried out in which pairs of disclination lines were inserted instead of twists. From the starting EC”‘, a “bare disclinated” D(EC’O’) was formed by applying to each spin a rotation by an anglef(x,J’), where (x, y, z) is the site of the spin (so the inserted disclinations always ran in the i direction in real space). (It is topologically impossible to insert just one disclination [ 6]!) The function f(x, y) varied by 360° around the line (x, y) = (3,3) and by 360” in the opposite sense around the line (x, y) = (9, 9). Note that f(x,-v) was not the analytic function appropriate to a vortex pair with periodic boundary conditions, but a simple approximation which had a discontinuity in its gradient at the boundaries, which must be smoothed out during the subsequent relaxation. The configuration D(EC”‘) was relaxed to EC (d’; this usually required more iterations (and more of the downward “steps” discussed in Section II) than the original generation of EC”’ did. The probable explanation for the longer relaxation is that disclinations have cores, and the position of the inserted disclinations was not adjusted to take advantage of the random spatial variations in the spin glass; consequently the disclinations must hop and annihilate in many places before reaching an equilibrium. Finally, I compared EC (O’ and ECcd’. The main difference from the procedure for the twist case is that here there is not much point to comparing D(EC”‘) and ECcd’; the disclinations usually move during relaxation, so such a comparison shows two pairs of disclination lines in many places. However, comparing with D(EC”‘) sometimes helped identify places where the disclination did not move (as in Fig. 8).

This insert-relax-and-compare procedure was repeated on 11 EC’s of 4 realizations, each starting EC (O’ being used once. These were the same independently generated EC(“‘s which were used for the correlation function calulations and for the twist- insertion experiments reported above.

I classified the results, much as in the twist case, according to whether the disclination pair “stayed.” In the three runs where the disclinations did “stay” the comparison of ECcd’ and D(EC”‘) had a projection of at least 0.5 and an (edited) disclination fraction of at most 2.4%; the projection being larger, and the disclination fraction smaller, than for the comparison of ECcd’ and EC”‘. In Fig. 8, I show the disclination lines from a typical run where disclinations “stayed.” (This figure was produced by comparison of configurations (D(EC (O’ ECcd’)) so the computer output , only showed the segments of the disclinations moved before relaxation; the location of the segments that stayed put, and the assignment of the other segments as “before” or “after,” was inferred from the comparison (EC , (d’ EC”‘).) There were also three runs in which the disclinations mostly annihilated and the remaining five runs showed confused results.

In the three cases in which the disclinations “stayed,” the energy difference was Ecd’ - E”’ = 1.8 f 5.4, /ECd’ - E”’ 1 = 3.9 f 3.4. Using, instead, the five cases in which both EC”’ and ECcd’ were nominally “converged,” the energy difference was Ecd’ -E(O) = 1.3 f 4.8, (E’d’ - E”’ 1 = 3.8 f 2.6. (For comparison, D(EC”‘), before relaxation, was about 300 units higher in energy than EC”‘.)


FIG. 8. Insertion and relaxation of a disclination pair. The disclinations originally inserted were essentially straight lines. The nominal disclinations after relaxation are shown by solid lines; arrows indicate where this was a change from the original position, shown by dotted lines. The dashed curves show three of the four small disclination loops which appeared during the relaxation; these represent noise in the comparison algorithms, or readjustments of small clusters. The circle shows a place where the disclinations are only one lattice unit apart; after more time it is possible they would begin to annihilate there.

C. Importance of “Reflected’ Domains

Inspecting the outputs from the twist and disclination comparison runs, one sees many regions of “wrong” sign in which the “order parameter” Q has undergone a reflection (in spin space) during the relaxation. They tend to have irregular, stringy, ill-defined edges: the value of (d3)ir the sign of which indicates whether Qi has undergone a reflection, changes gradually from strongly negative, in the middle of the reflected domain, to strongly positive outside. I analyzed some runs by hand, making a (somewhat subjective) operational definition of a “cluster”: wrong-sign sites which were connected were grouped into the same “cluster,” except that these were cut at “bridges” with only -4 spins cross section, and occasionally large wrong-spin domains which had two separated cores of strong reflection were divided. Clusters of less than about 20 spins, which never had a strong-reflection core, were ignored. I typically found 50-200 spins per cluster.

Every stable disclination line or loop seems to be associated with reflected regions (e.g., see Fig. 6) whereas not every reflected region can be sensibly interpreted in terms of disclination lines. Therefore, I speculate that reflected clusters (whose boundaries correspond to the “reflection wall” topological defect discussed in the preceding paper [ 61) are in some sense the more fundamental defect. Independently, very recent simulations by Walstedt [ 191, in which low-energy paths between closely related EC’s were found, suggest that the difference is the reflection of a cluster.


Perhaps the only way defects can be stabilized against (local) untwisting is by the barrier which always exists between a (local) configuration and its reflection.

The value of (d3)i often changes its sign quite gradually as one leaves a reflected region and it is nearly zero in many places. A small (d3)i value means we can describe the comparison of the EC’s as follows: we project the spins from the respective configurations into respective two-dimensional subspaces; the overall rotation consists of a part which brings the subspace planes into agreement plus a rotation of the two components within the subspace. The respective third components, normal to the respective subspaces, are parallel on about half the sites within the averaging volume used to define Qi, and antiparallel (i.e., show a relative reflection in the subspace plane) on the other half of the sites. This situation corresponds to the “flattening” discussed in Section IV of the preceding paper [6], and tends to occur on the boundaries of the reflected domains.

One could further conjecture that, in fact, all the third spin components of both configurations are nearly zero in the “flattened” region; i.e., these are places where the configuration naturally happens to be XY-like. A comparison of the configuration with itself would show small (d3)i values locally. It is very plausible that walls would be “pinned” in such places: imposing a reflection there need not alter the spins at all, so this way of patching together reflected and unreflected regions should have a low energy cost. If correct, this conjecture would help explain why the spin angles appear to vary smoothly even on the “wrong-sign” side of a reflection wall (see Fig. 2) where they are, in principle, meaningless. There is not sufficient data to test the conjecture.

It is clear that there must be reflected domains in comparisons of independently generated EC’s, but it is conceivable that the reflected clusters which appear during relaxation of twists and disclination pairs might be artifacts, caused by large angle variations; perhaps all the physically meaningful defects are expressed by proper rotations. As motivation, consider the comparison of an EC with the bare-twisted T(EC); from Eq. (5.2), the singular values are dy = ($, 0.0) + O(N- ‘I’). In a finite system the last two singular values will deviate randomly from zero so that half the - - time, when d,d, < 0, the decomposition procedure will produce a reflected R. Similar effects could occur with the local Qi when its averaging volume (which includes 27 spins) spans large variations in rotation angle due to a disclination line.

Some such mechanism could well account for many cases of weak@ negative (d3)i values. However, there is some evidence against the mechanism I suggested: (i) comparison of EC and a bare-disclinated D(EC) shows little spurious reflection, and (ii) the reflected regions in the comparison of EC”’ to EC(‘) are found to be almost the same as in the comparison of EC”’ to T(EC”‘); if they were an artifact of large gradients we would expect them to be quite sensitive (as the nominal disclination positions are!) to the subtraction of the uniform twist gradient that we are in effect making when we compare to T(EC”‘). In any case, it would be hard to explain as spurious the strongly reflected regions where (d3)i is nearly as negative as it can get, i.e., the optimal projection Pop, = d, + d, - d, is near its upper limit of unity. Around these places the configurations being compared are certainly related by a nearly rigid rotation of the “wrong” sign.


I have now asked both “are disclinations spurious effects of reflections” (part A) and vice versa (just above), and answered “no” in each case. As a final judgment on the small defects nucleated during the relaxation of twists and disclinations, I will note that their dimensions are of the same size as their “cores” (estimating the core diameter to be at least two lattice constants+isually the disclination loops do not really have a clear space through their centers where the rotation can be clearly defined (the loops in Fig. 4 are unusually open). Under these circumstances there is a sort of uncertainty principle that it is impossible to identify a defect as either a disclination loop or a reflection. It is best to use “complementary” language: the defect is both a disclination loop and a reflection at the same time.

The conclusion of this section is that it is possible to create, by insertion-and- relaxation, stable EC’s which can be identified as having twists or disclinations (relative to another EC); however, this is not the most common outcome, due to the large gradients imposed. Many rearrangements which occur during the relaxation can be understood in terms of defects, though the details are murky. The average energy change after insertion and relaxation is zero within my error bars, i.e. (in the case of twists), less than about 1 unit. This result is consistent with that of Reed [ 11, who says the “excitation energies” are “very small ” (0.2-1.0 units in his Fig. 1). It would suggests that the inserted-and-relaxed EC’s, even where the inserted object clearly “stayed,” are not distinguishable from the starting EC”“s (for instance, I would conjecture that they have the same probability as the EC”“s of being reached by relaxation from random initial conditions). This could be tested by use of the absolute measure of twist (5.1). Groups of spins which undergo a reflection relative to their starting orientations seem to be an important ingredient in describing the relation between the starting and final EC’s, and perhaps between all neighboring pairs of EC’s.

VI. QUASI ONE- AND TWO-DIMENSIONAL SAMPLES

The third category of experiments used samples which were very narrow in one or two of their dimensions: rod-shaped 4 x 4 x 16 samples (N= 256) and slab-shaped 12 x 12 x 5 samples (N = 720). I still use periodic boundary conditions in all directions. Since the “order parameter” correlation length <oT is at least one-half the narrow dimension, one expects correlations in that direction to be enhanced since one spin “sees” another in two or four directions. They are indeed enhanced, especially for the rod-shaped samples, where the Q matrices were all practically the same within each 16-spin layer (see Fig. 10).

For several realizations of each shape, independent EC’s were generated from random intial conditions by the algorithm described in Section II (with the same stopping criterion D’ = lo-‘; the average number of iterations was about 700). All possible pairs of EC’s belonging to the same realization were then compared by the


algorithms described in Section III. I plotted out the projected rotation angles $i, in order to look for twists, and the third singular values {&} from the decomposition of the Q matrices, in order to look for reflection domain walls, and analyzed the results by hand.

The projected rotation angles were calculated as described in Section III. We do not know in advance which axis a^ the rotations should be about, so we choose a^ to be the eigenvector of q corresponding to the largest singular value. One can see this is reasonable by considering (5.2): in the case of a bare twist this eigenvector is the original twist axis. We also do not know in advance which sign to specify for the rotation matrices; on the average the best choice is positive on half the sites and negative on the other half. Therefore, in many of the realizations comparisons were run once for positive and once for negative sign, since these give different results for the projected angles and the disclination locations.

A. Slab-Shaped Samples

Four slab-shaped realizations were produced and a total of 15 EC’s were generated for them, making 21 pairs fr comparisons. The average energy/spin was -1.84. For the slab-shaped samples, only the 12 X 12 array of column-averaged rotation matrices and projected angles were used in the analysis: the rotation matrices were defined using weightings such that each of the five spins in a column contributed equally and the column-projected angles (b,,, were defined from these in close analogy with the layer-projected angles &,,, defined in Section V (but choosing the axis about which they are computed as explained just above). An example comparison is shown in Fig. 9. The nominal disclinations are not shown since they are numerous (about 20 piercing the slab, in a typical comparison) and probable spurious (most are in wrong-sign regions).

There are clearly clusters of sites with positive and negative signs, like the domains of Ref. [6, Sect, IV]. I grouped sites into domains, using the output from the column average rotation matrices, by two rules: (i) the boundary between two regions of opposite sign is a domain wall (if a cluster of less than -20 spins becomes isolated, it is treated as if there were no spins there ), and (ii) if two parts of a region of the same sign could be separated by putting a wall between only - 10 or fewer nearest-neighbor sites, this was done (except when this would have isolated a cluster of less than -20 spins). The average domain size was -135 spins (geometric mean, because the size varies widely); the were four or so domains in each comparison on average.

In no case was anything like a twist observed: indeed, a twist can only be defined in a region where the Qi matrices have the same sign. All the disclination lines observed were in regions where we were forcing a fit to a rotation with the “wrong” sign. In cases where domains of one sign percolated (in the long directions) across the sample, I did find some cases where the projected rotation looks like a twist. In one of these cases two paths exist, one of which clearly has a twist and one of which does not (Fig. 9). Thus, we have a sort of disclination but with a core in the sizeable region of “wrong” sign between the paths. This is suggestive for picturing EC’s in


(b)

FIG. 9. Projected angles @,,, relating two independently generated EC’s of the same realization of a 5 x 12 x 12 sample of spin glass. Each arrow represents a rotation matrix Rcc,, derived from summing over a column of five spins. The signs of the R,,, matrices were constrained to be (a) positive or (b) negative. The “wrong-sign” regions are shaded: locations of the nominal disclinations are not shown. In Fig. 9b. a twist is clearly visible along the path A’ while there is clearly no twist along the path BB’.

general: having the core in a reflected region means that the singularity occurs in a place where it is meaningless and costs no energy. This is reminiscent of the pinning of vortices by impurities by superconductors- the core is attracted to positions where the order parameter must be zero anyway.

B. Rod-Shaped Samples

Three rod-shaped realizations was produced and four EC’s were generated for each; however, in one realization two EC’s were identical (modulo O(3)), so there


FIG. 10. Projected angles Qi relating two independently generated EC’s of the same realization of a 4 X 4 x 16 sample of spin glass, showing one 4 x 16 layer using the same conventions as in Figs. 2 and 6. The strong correlations within each 4 x 4 layer are visible in the almost identical orientations in each column of the figure. This pair of EC’s shows a clear relative twist; there are no disclinations.

were 11 distinct EC’s. Their average energy per spin was -1.79 units. Of the 1.5 distinct pairs of EC’s, about 5 appeared to have a relative “twist” (the projected angle icreased monotonically along the length of the rod). One of these is shown in Fig. 10. Also, 3 comparisons showed zero twist, and the other 7 could not be described by proper rotations at all. Note that the rotation matrix often varies continuously through a band of layers of the “wrong” sign. In fact a rotation is uniquely defined from just two spin components provided we specify a given sign, so these bands may well be places where the spin configuration is XY-like, as discussed in the preceding section.

There were unmistakable reflected domains: bands of reflected and unreflected spins, alternating along the “rod,” some as short as two layers. A comparison of two EC’s most typically included two or four domains; the average size (geometric mean) of a domain was about 70 spins. This is exactly what we would see if each sample were composed of 6 independent clusters each of which had two states (reflected/unreflected&-one assumes the overall twist adjusts to satisfy the bonds across the walls. When two neighboring clusters had the same sign, they would be counted as the same domain by my operational definition. This independent-cluster model predicts that a total of Nmin = 32 distinct EC’s exist (modulo O(3)).

Say we generate n EC’s from random initial conditions, and these include a total of n’ EC’s which are distinct (modulo O(3)); then (n’) z N,,,(l - exp(-n/N,J) As long as n’ < n, one can obtain a crude estimate of N,i, without having to generate all the EC’s. For the rodlike samples n = 4, and n’ = 3 in one realization (in another realization, one incompletely converged EC may be converging to another EC); this implies LV,,,~, z 10-25, so the six clusters are not completely independent.

VII. SUMMARY

To summarize the results of the preceding sections: I have described the typical behavior of the Walker-Walstedt relaxation procedure, previously unreported, and exposed its relation to zero-temperature relaxational dynamics and to “white averaging.” Pairs of spin configurations were successfully reduced to comigurations of rotation matrices, confirming the validity of a slowly varying “order parameter”


and rigid rotation of the spins within local clusters. The rotation matrix has a longer correlation length than anything else in the spin glass.

I have established that disclination lines and twists exist, but it is still questionable whether they are useful for describing spin glass EC?. As discussed in Section V, I believe that domains in which the spins undergo a reflection are important; they seem to be associated with all stable defects, and the system does seem to be rather “soft” to the creation of reflection walls. However, I cannot say as such about reflected domains as I can about twists and disclinations, because my procedures were chosen to look for the latter objects. In any case, a sort of uncertainty principle prevents one from clearly classifying most defects, since the distance between them is not much larger than their core size. As for the defect “excitation energies,” they are small- perhaps the order of one unit (Sections II and V).

There are many estimates, crude but independent, of the “patch size” N,, that is, the number of spins corresponding to one degree of freedom in constructing EC’s [6]. Most of these seem to be consistent with N, z lo&150 spins for Heisenberg spin glasses, independent of whether the interactions are nearest-neighbor, RKKY, or infinite-range. There are three ways to estimate N,:

(i) From the number of minima Nmi, as a function of the number of spins N, one derives the entropy-of-minima g$, = log(N,,,)/N for N -+ 00; then N, % g$, . The analytic estimate of Bray and Moore [ 151 for number of minima of the inlinite- range model is g;& 2 125; I inferred [6] g;:, z 90-l 10 from Walker and Walstedt’s RKKY spin glass simulation results [20]. I did not derive any estimate of g,,, for the Edwards-Anderson spin glass from the simulations reported here.

(ii) From N, z xor, the correlation volume of the transverse order parameter correlation function. I found (Section I, above) xaT E 260.

(iii) The typical size of the blocks of spins where closely related EC’s differ gives another measure of N,. Reed [ 1 ] found that a block of at least 53 spins needed to be rotated in order that subsequent relaxation would lead to a new minimum, i.e., N, E 125. In the simulations reported here, I grouped “reflected” spins into clusters and counted spins per cluster; the geometric means were N, g 130 for 12 x 12 x 12 samples (Section V), N, z 135 for 12 x 12 x 5 samples, and N, g 70 for 16 x 4 x 4 samples (Section VI).

The results suggest interesting possibilities for further simulations. The comparisons of independently generated EC’s should be repeating using EC’s generated by some sort of annealing process, which will selectively converge to the local minima with the lowest energies [20a]. As argued in Section VI of the preceding paper [6], the defects will then be fewer and farther apart so that it may become easier to identify them in terms of the topological classification in Section III of that paper.

My speculation that all “defects” or “two-level systems” in spin glasses can be described by reflections should be tested. This requires detailed study of the nature of the barriers between two configurations, as has been begun by Walstedt ( 191. It is

SEARCH FORDEFECTS IN A SPIN GLASS 357

interesting to note that even in small systems (NE loo), which have a unique minimum modulo O(3), the minima which are related by a reflection are separated by a barrier, which should be easy to study.

APPENDIX A: LINEARIZED THEORY OF RELAXATION TO EC’s

This appendix gather some results on the behavior of the Walker-Walstedt minimization algorithm when the spin configuration is very near a local minimum (an EC) {s?}, so that its valley can be represented by a quadratic form in the deviations from the EC. I consider m-component spins (m > 1) for generality.

I. Energy for Small Deviations

Expanding the Hamiltonian (1.1) about an EC, one finds

E = -E, + 4 s hi&)2 - 5 Jij6$ . 6& i ii

(A.11

where hi = j li 1, the exchange field on spin i, and where the spin deviation 6<. = s’- ~7. Note & I q since the spins are unit length.

I will now introduce some formalism which will streamline the expression of (A. 1). For each $j we choose [9] an orthonormal basis je^,,, p= l,..., m - 1) for the subspace of deviations, choosing the signs so that {ST, ti., ,..., e^,,,-, 1 is a right-handed orthonormal basis, and then express

If we have XY spins (m = 2) and represent them by angles Bi in the XY-plane, then 68, as defined by (A.2) is just 19~ - 8:. We can consider e, as an m X (m - 1) matrix: then 6fei = (6,,), the (m - 1) x (m - 1) identity matrix, and e,ef = (a,, - s~~s$), the (m X m) projection operator onto the subspace of deviations. Now we define projected couplings jiO,jO = (eTej),, Jij; this a square symmetric matrix which should be pictured as N x N blocks which are each (m - 1) X (m - 1) matrices. For XY spins it reduces to jii = J, cos(19~ - ej). Similarly we define Hip,jo = /~~6~~6,,, a block diagonal matrix with each block a multiple of the (m - 1) x (m - 1) unit matrix. Finally, we write the deviations as a vector (se,,, ,68,*, ,..., &9,,(,- ,, ,68,, ,..., 61!&,.,- ,) and in this notation the energy is

E=E,+f&E68, (A.3)

where E=H-3 (A.4)

is the Hessian matrix, the inverse of the susceptibility matrix. It is always nonnegative definite, since we are at an energy minimum. There are always


m(m - 1)/2 zero modes corresponding to uniform rotations of all the spins. In the XY case the uniform eigenvector is just N-“2( 1, l,..., 1). I assume all the other eigenvalues {A} are positive, although it is well established that they can be arbitrarily small [3, 131.

2. Convergence Criteria and T = 0 Relaxational Dynamics

It would be very convenient if we could evaluate E -E, without knowing the final relaxed configuration, for two purposes: (1) as a measure of convergence, and (2) so that, given the energy E after a finite number of iterations, we could estimate E,, which in some calculations (e.g., spin-stiffness constant) is the principal object. This is not possible, but the two quantities defined in (2.1)-(2.2), which I (and others) have used as convergence criteria, are related to the energy. With some work one can reduce these to quadratic forms like that of the energy:

ND = 68 D 60, where D = EH ~ *E, (A.3

Nw=6ewde, where W = EH-‘E, (A4

Then (NW), = NT = (E, - E& in thermal equilibrium. If one makes the (natural, but false) hypothesis that the slow-relaxing modes exchange energy rapidly (because of the nonlinearities omitted from (A.l)), then one expects a quasi-equilibrium, and it is natural to use W to estimate E,.

To see the true behavior of E, D, and W after many iterations, let us decompose 68 in terms of the eigenvectors of E, with dn being the component in the direction (in (m - 1) N-dimensional space) of the eigenvector with eigenvalue ;1; then

E= f ;A#:, (A.7a)

(A.7b)

Here (...)1 means an average over sites, weighted by the squared amplitude of eigenvector L on the sites. There should actually be cross terms in (A.7b) and (A.7c) proportional to dn#U but their coefficient, the matrix element of hi2 (resp. h; ‘) between the eigenvectors of 2 and ,u, is of O(N-I’*) and of random sign, so we neglect them.

The third quantity used as a convergence criterion is / dE/dt/. To evaluate it, I assume a simple zero-temperature relaxational dynamics, the same as used (for example) by [ 131:

ds’Jdt = --y(g) I, (A-8)


where (h7:), means the projection of the instantaneous exchange field perpendicular to Zi. Linearizing, as before, near the EC,

d(&)/dt = --y(H - 3) 68, (A-9)

ldE/dtI = y~L29:. .\

(A. 10)

One should keep in mind that the large-A components will relax very rapidly so that after many iterations only the small-1 behavior matters. Now, one expects that the large-A eigenvectors tend to be localized on a single site i, with d r hi. There are many small-l eigenvectors, but there are no hi’s near zero (see Fig. l), so these eigenvectors must be fairly extended, as indeed found by Grzonka and Moore [3] for XY spins. In this case one can replace (.VV)A by an average (.*.)i with equal weights on all sites, and factor it out of the sum. The consequence is that all three convergence measures-D, W, and dEfdt-are proportional, with ND z (h,:2)i IdE/dt 1 and NW g (hL:‘)i ldE/dtl. This was observed in simulations, and the ratios had the predicted values. Thus W is not a good estimate of E -E,.

According to Ref. [ 131, the density of eigenvalues p(A) goes as A” near I = 0 for XY spins, with x = 0 (d = 2) or x = $ (d = 3). Since our random starting configuration has (4:) ; 1, this means (compare Ref. [ 131) that

E-&-t-“-’ and D, W, ldE/dtl- t--“’ (A.1 1)

at large times; this would allow estimation of E, after t iterations within an error much smaller than E(t) - E,, where brute force might require hundreds more iterations to get E(t) z E, within the error. Unfortunately, the relaxations reported here rarely reach the quadratic valley where (A. 11) is valid.

3. Walker- Walstedt Relaxational Dynamics

Now I will discuss the actual relaxation algorithm used (described in Section II) and its relation to the relaxation dynamics of (A.8).

The Walker-Walstedt algorithm means that in each spin update, we minimize the energy with respect to just the (m - 1) degrees of freedom of spin k, holding the others fixed. In the linearized notation developed above, we set

se; = hi’ \’ 3, sfJj. ,?k

(A.12)

Note that (A. 12) is a linear operation on the Gets, and so we can represent it by a matrix:

where

68’ = B’k’d8 = (I - hi ‘Pck)E) 60, (A.13)

Pi,“! jO = Sik Sjk 6,, . (A.14)


Here 1 means the identity matrix and PCk) is just the projection operator which projects to the (m - 1) x (m - 1) block associated with spin k, so that

L7 h; lpck)E = c P’k’H - kE r-i

k k

= H-‘E. (A.15)

Finally, a single relaxation iteration is expressed by the product of single-spin updates

&I’ = B&I, B = B(N)B(N-l)(...) B(*)BC~). (A.16)

Let us note two things about B. First, it is not exp(-yE) which (if y is small) is the matrix which corresponds to the dynamics (A.8). Second, B is not symmetric. Therefore, its eigenvalues may have imaginary parts, leading to spiraling behavior in XI-space. This can include oscillations in the magnitude of the displacement, II6811 : the energy always decreases (i.e., E - BTEB is positive definite) but //&I//* need not decrease (i.e., 1 - BTB is not necessarily positive definite). If I(StI/(2 oscillates, one presumes D’ may also.

Let us consider two tractable updating procedures (of which neither is actually the Walker-Walstedt procedure). First procedure: update in parallel, i.e., compute all the 6’s and then update all the spins at once; this is described by

g=H-‘j

= (1 - H-‘E). (A.17)

Since H-‘E is similar to the symmetric (in fact, nonnegative definite) matrix H-“*EH-“*, its eigenvalues are real (in fact, nonnegative) and so those of (A. 17) are also real (and, in fact, less than 1).

Second procedure: choose randomly and independently which spin to change in each update. (This is not as efficient as using a fixed permutation of the spins; for instance, if we choose the same spin twice in a row, then the second update will not change it at all.) Then one easily derives the expectation after N updates

= (1 ---‘H-‘E)N (A.18)

so (B) z exp(-H-‘E). In the subspace of small-eigenvalue modes H ’ g (hi ‘)i 1 so that this kind of updating dynamics is (A.9) with y = (hl:‘)i. Furthermore, the eigenvalues of (A. 18) are real and less than 1, for the same reason that those of (A. 17) are. Unfortunately, what actually determines the relaxation is (log B) and this is not the same as log(B). The best justification one can give that the eigenvalues of (A.16) should be nearly real in practice is that the actual (Walker-Walstedt) updating procedure is intermediate between (A. 17) and (A. 18).

SEARCH FOR DEFECTS IN A SPIN GLASS

APPENDIX B: VALUES OF RANDOM Q MATRICES

361

The purpose of this appendix is to derive some results for the distribution of singular values (0,) of the Q matrices arising from comparisons of spin configurations, using random matrix theory. For didactic reasons, I do this first for the comparison of a spin configuration with itself (Q”*U”‘), and then repeat the derivations for the somewhat more interesting case of the comparison of two different configurations. For generality, I will do the computation for the case of m-component (unit) spins.

Each time, I start by deriving the appropriate ensemble of Q’s arising from comparisons of completely random spins. The interesting case is when there are correlations between the spins. One can get the Q distribution in this case, too, given the appropriate correlation function (C,(F) or Co,(?)). The local averaging which was used in defining a local order parameter Qi can easily be dealt with. Once one knows the distribution of the Q’s, the distributions and averages of the singular values follow from random matrix theory.

l.Q . C1’l’* Random, Uncorrelated spins

I am starting with the Q”,” matrices from a comparison within the same configuration of unit Heisenberg spins, {$I’}. Consider the local, unaveraged

Qf;;' = s~;)s$‘; P.1)

the averaged Q matrices (local Qi, overall Q) are linear combinations of (B.l). (Since I am dealing with only one configuration, I will leave off the superscripts (1) and (1, 1) until part 3 of the Appendix.) Now I assume the spins are random and uncorrelated; then

Also note that

Writing

CSiaSiBSLySi8) = m(ml+ *) @,,$a + f&J46 + L43J*

we find that

(B-2)

(B3)

P.4)

P.5)


The overall Q matrix is

a sum of many terms. Note that the singular values of Q are the same as its eigenvalues, since it is symmetric; and its eigenvalues will be given by

Dy=i+ CSD,, m 03.7)

where {6D,} are the eigenvalues of 60. Now,

03.8)

with

2

‘2=m(m+2)N’ (B.9)

i.e., the m(m - 1)/2 independent off-diagonal elements S& _= SOD, have variance a2/2 and the m diagonal elements S&, obey (@a,~Q,,) = (6,, - m-‘) a2. Furthermore, since So is a sum of many independent terms of finite variance, its components obey a Gaussian distribution, which is completely determined by the variances (B.8). It can be conveniently expressed as

P(@) a e- Tr(d@9r)/202&Tr @). (B. 10)

This means So is distributed according to the “symmetric orthogonal ensemble” [21], except that in addition SQ is restricted to be traceless (which follows from N-‘-p&E 1 exactly). The eigenvalue distribution for matrices in the symmetric orthogonal ensemble is derived in, e.g., Ref. [2 1 ] ; a minor modification gives the distribution for the traceless symmetric orthogonal ensemble,

P@D, , 6D, ,..., SD,) a n IdD, - 6D,1 eCz:y(sD?‘2028 (B.11) a<5

If we have short-range correlations between the @i’s, we will still find the results (B.8)-(B.l l), but with changed values of u2. Even with random uncorrelated spins, we have correlations between the the locally averaged Q”“’ matrices

iji = c wijQj (B.12)


(same as Eq. (3.2)), where the weighting function is translationally invariant (depends only on the vector from site i to site j) and normalized by cj Wij = 1. Thus (pi) = m-‘1 as before but there are now correlations between 0;s due to the averaging:

using (B.5). Note how the prefactor is a convolution of the weighting function with itself. With the weighting function I used for simulations this convolution is zero for displacements beyond cj = [222]. In particular, if we consider one site i, we see that SGi is distributed like (B.8) and (B.lO) but with

2

u2 = m(m + 2) 7 \’ w;k (B.14)

in place of (B.9). For the weighting function I used, Ck w:~ = 2715 12. Given u2, one can derive the expectations, variances, and cross-correlations of the

DY’s from Eq. (B. 11) by elementary integrations. In the Heisenberg case one finds:

(0,) = f + 1.47a,

(D2) = 4, (B.15)

(D,) = f - 1.470.

The matrix of cross-correlations, with elements (D,D,) - (Dn)(Dn), is

0.268 -0.083 -0.185 -0.083 0.167 0.083 u2.

i (B.16)

-0.185 0.083 0.268

2. Q (I “* Random, Spins, with Correlations .

Now I will apply the same method to the case of configurations which are actual EC’s. As with the locally averaged Q’s (Eqs. (B. 12k(B.14)), there are correlations between different Qi)s, but now this is intrinsic to the spins, not an artifact of averaging. Using the correlation function C,(F) as input and otherwise assuming the spins are random, I will again be able to calculate the average values of the d;s.

Because the probability distribution of Jij is symmetric about zero, we can assume that (B.2) still holds; however, (B.3) is replaced by

(B.17)


which follows from the symmetry under a t+ p, y c-) 6, from (Isit2 ISjl’) = 1, and from the definition of the correlation function,

C,(Fjj) = (m(($ * ?J”) - l)/(m - 1). (B. 18)

Equation (B.5) is correctly generalized if we replace 6,-t C,Lcj). Then Eqs. (B.8), (B. lo), and (B.11) again describe the distribution of the total SQ matrix, provided we take

u2 = 2&/m(m + 2) N, (B. 19)

where xS = E C,(c7) (independent of i by translation invariance). If we wish to substitute the C,(F) appropriate to an infinite system into (B.19) then of course we must make sure that each dimension of the N-spin system is much larger than the correlation length <, .

Finally, one can find the expected values of the (dJi’s when one both uses the correlated spins from an EC (ci} and does the local averaging; the details can be left to the reader.

3.Q . (132’* Two Uncorrelated Configurations of Random Spins

In the rest of the Appendix, I will repeat all of the above calculations for the case where we are comparing two different configurations, {q”} and {?‘j2)). Of course, (sj;‘sj;‘) = 0, so

(@) = (0) = 0. (B.20)

(Here and below, the configuration superscripts are assumed to be (1, 2) on all Q matrices.) Consequently,

and

a, e,> = L4d-J2~ (B.22)

where

u2 = l/m2N. (B.23)

The components of Q, like those of @ in the first part of the Appendix, obey a Gaussian distribution, but it is simpler than (B.lO) since the components are independent and all have the same variance u2. In place of (B.lO), we have

JYQ) a e -Tr(QQT)/2uZ (B.24)


This is called the “general orthogonal” matrix ensemble since the matrices are not restricted to be symmetric. The distribution of singular values can be evaluated by a slight generalization of the method in Mehta [21]. Briefly, one writes Q = R, diag(D,) Ri, changing from the m* independent variables Q,, in (B.24) to the m singular values {D,}, the m(m - 1)/2 angles representing the orthogonal matrix R,, and m(m - 1)/2 more for R,. All one needs is to compute the Jacobian matrix of this transformation. First make a change of variables in (B.24) to the m diagonal elements Q,, , and the off-diagonal symmetric and antisymmetric parts,

i(Q,, f Q,,> Mm - W variables each). It turns out that the Jacobian matrix from these variables can be factored into a part depending only on the angle variables times a diagonal matrix with elements 1 (m times). and (D, f D,) for all a < /3. The resulting distribution of singular values is

(B.25)

One could also deduce this from a result of Dyson (Ref. [22, Eq. (1.4)]) for the eigenvalues of a matrix H = CL, (Q(r)TQ(“‘), where each QCr) obeys the distribution (B.24); taking k = 1 and noting that the eigenvalues are just (Di}, his Eq. (1.4) for the eigenvalue distribution is identical to (B.25).

(I.*’ In the case of the local averaged Qi defined again by (B.12), one can derive the analog of (B.13) using (B.21). The result ‘is that Qi is distributed like Q in (B.22) but with

(B.26)

From Eq. (B.25) one can again derive the expectations, variances, and cross- correlations of the Dy’s, as a function of d*, by elementary integrations. Averaging with (B.25) over all Dy’s one obviously has (D,) = 0; this reflects the fact that for any given matrix in the ensemble, I have integrated over all the trivial sign changes and permutations (see Section III above and Section II of the preceding paper). However, my algorithm takes the {D,} positive and sorts them into descending order; one can implement this by restricting the integration with weight (B.25) to D, > D, > D, > 0. I have evaluated these integrals numerically in the case m = 3, finding:

(Dl) = 2.520,

(D,) = 1.330,

(D3) = 0.4 10.

The matrix of cross-correlations, with elements (D, D,) = (D,)(D,), is

! 0.404 0.120 0.030 0.120 0.233 0.052 0.030 0.052 0.100

(B.27)

(B.28)


4.Q . (1q2’* Two Configurations Correlated like EC’s

The distribution of Q matrices for comparisons of two EC’s is the most interesting one; it is calculated by straight analogy with part 2. The analog of (B. 17) is

(B.29)

It follows that Q obeys (B.22) and (B.24), but with

where x0 = cj C,(i$).

o2 = xo/m2N, (B.30)

In making this calculation, I have implicitly assumed that there is no true long- range order in the system defined by my ensemble of Q’s. This is certainly true for the “white average” induced by the Walker-Walstedt relaxation algorithm (Section II); EC’s of all energies are weighted equally, so within the set of EC’s it is like being at infinite temperature. I will only note that (B.22) with (B.30) would also be valid for the case of long-range order, in which case xc/N -+ (const) as N + co. However, it is not clear to me whether the distribution must be Gaussian, (B.24), since that depended on the system containing a large number of correlation volumes.

ACKNOWLEDGMENTS

I am grateful for useful discussions with Scott Kirkpatrick. Michael Moore, and B. I. Halperin (who suggested and guided this project). I am indebted to B. I. Halperin and T. C. Halsey for detailed comments on the manuscript. Chandan Dasgupta helpfully provided a relaxation program and some EC’s of an RKKY spin glass in the preliminary stages. This work was supported by NSF Grant DMR824743 1.

REFERENCES

1. P. REED, J. Phys. C 12 (1979), L859.

2. F. D. M. HALDANE, C. JAYAPRAKASH, S. KIRKPATRICK, AND J. VANNIMENUS, Bull. Amer. Phys. Sot. 25 (1980), 209; and personal communication (J. Vannimenus).

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4. 1. MORGENSTERN AND H. HORNER, Phys. Rev. B 25 (1982), 504. 5. F. BARAHONA, R. MAYNARD, R. RAMMAL AND J. P. UHRY. J. Phys. A 15 (1982). 673.

6. C. L. HENLEY, Ann. Physics 156 (1984), OOC&OOO.

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computer search for defects in a d = 3 heisenberg spin glass · some of my approach was inspired by...

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